Scatter Diagram

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Scatter Diagram

• a plot of paired data to determine or show a
  relationship between two variables

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Scatter Diagram
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Linear Correlation

• The general trend of the points seems to follow a
  straight line segment.

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Linear Correlation
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Non-Linear Correlation
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No Linear Correlation
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High Linear Correlation
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Low Linear Correlation
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Correlation Coefficient, r

• The correlation coefficient is a number that
  indicates the strength of a linear relationship
  between two variables, x and y.
• -1ltrlt1
• If r1, there is a perfect positive linear
  correlation.
• If r0, there is no linear correlation.
• If r-1, there is a perfect negative linear
  correlation.
• The closer r is to 1 or -1, the better a line
  describes the relationship between the two
  variables x and y.

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Positive Correlation
y
x
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Negative Correlation
y
x
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Computation formula for r

• Let nsample size. Then

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Example
Let x be a random variable representing wind
velocity and let y be a random variable
representing drift rate of sand.
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Example
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Computation
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Coefficient of Determination

• a measure of the proportion of the variation in y
  that is explained by the regression line using x
  as the predicting variable

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Formula for Coefficient of Determination
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Interpretation of r2

• If r 0.949, then what percent of the variation
  in minutes (y) is explained by the linear
  relationship with x, miles traveled?
• What percent is explained by other causes?

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Interpretation of r2

• If r 0.949, then r2 .900601
• Approximately 90 percent of the variation in
  drift rate (y) is explained by the linear
  relationship with x, wind velocity.
• Less than ten percent is explained by other
  causes.

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Warning

• The correlation coefficient ( r) measures the
  strength of the relationship between two
  variables.
• Just because two variables are related does not
  imply that there is a cause-and-effect
  relationship between them.

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Questions Arising

• Can we find a relationship
  between x and y?
• How strong is the relationship?

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When there appears to be a linear relationship
between x and y

• attempt to fit a line to the scatter diagram.

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When using x values to predict y values

• Call x the explanatory variable
• Call y the response variable
• A lurking variable is a variable that is neither
  an explanatory or response variable. Yet, a
  lurking variable may be responsible for changes
  in both x and y.

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The Least Squares Line

• The sum of the squares of the vertical distances
  from the points to the line is made as small as
  possible.

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Least Squares Criterion
The sum of the squares of the vertical distances
from the points to the line is made as small as
possible.
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Equation of the Least Squares Line

• y a bx

a the y-intercept
b the slope
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Finding the slope
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Finding the y-intercept
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Find the Least Squares Line
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Finding the slope
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Finding the y-intercept
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The equation of the least squares line is

• y a bx
• y 2.8 1.7x

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The following point will always be on the least
squares line
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Graphing the least squares line

• Using two values in the range of x, compute two
  corresponding y values.
• Plot these points.
• Join the points with a straight line.

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Graphing y 30.9 1.7x

• Use (8.3, 16.9)
• (average of the xs, the average of the ys)
• Try x 5.
• Compute y y 2.8 1.7(5) 11.3

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Sketching the Line Using the Points (8.3, 16.9)
and (5, 11.3)
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Using the Equation of the Least Squares Line to
Make Predictions

• Choose a value for x (within the range of x
  values).
• Substitute the selected x in the least squares
  equation.
• Determine corresponding value of y.

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Predict the time to make a trip of 14 miles

• Equation of least squares line
• y 2.8 1.7x
• Substitute x 14
• y 2.8 1.7 (14)
• y 26.6
• According to the least squares equation, a trip
  of 14 miles would take 26.6 minutes.

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Interpolation

• Using the least squares line to predict y values
  for x values that fall between the points in the
  scatter diagram

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Extrapolation

• Prediction beyond the range of observations